Theory

One typical implementation of the SAU is the one proposed by Jensen.^[9] In his model, only one array element in the transducer is used each time to emit a nearly spherical wave that will cover the whole region of interest. All elements are used to receive the back-scattered signal at the same time, obtaining a low-resolution for each firing. The back-scattered signal contains information required to form the image in all directions and the specific direction can be acquired by applying different delays to signals received by different elements, thus this process works as dynamic focusing at the receiver. By summing all the low-resolution images acquired each time with a determined weight, the final high resolution image is formed and dynamic focusing at transmission is synthesized.^[9]

Equations representations

Assuming the lth focal line in the low-resolution image acquired from the ith element's firing is represented as:

${\displaystyle L_{l}(t;i)=\sum _{j=1}^{N}a_{l}(j)r[t-\tau _{l}(t;i,j)]}$

Here, ${\displaystyle a_{l}(j)}$ represents the jth receive element's apodization factor. ${\displaystyle r(t)}$ represents the received signal and ${\displaystyle \tau _{l}(t;i,j)}$ represents the delay applied to the jth receive element when the element receives signal transmitted from ith element, to beamform at a specific direction. So the low resolution image can be represented by

${\displaystyle \mathbf {L} (t;i)={\begin{bmatrix}L_{1}(t_{0};i)&L_{2}(t_{0};i)&..&L_{S}(t_{0};i)\\L_{1}(t_{1};i)&L_{2}(t_{1};i)&..&L_{S}(t_{1};i)\\.&.&..&.\\.&.&..&.\\L_{1}(t_{h};i)&L_{2}(t_{h};i)&..&L_{S}(t_{h};i)\end{bmatrix}}}$

By summing up all the low-resolution images, the region of interest for the high resolution image can be represented as

${\displaystyle \mathbf {H} (t)=\sum _{i=1}^{N}\mathbf {L} (t;i)}$

The high resolution image is dynamically focused both at the transmission end and the receive end.

Imaging efficiency

The major concern for the medical ultrasound imaging is the frame rate, which is determined by the number of lines that are formed by the focal points required to reconstruct the image, and the pulse repletion frequency. By considering these two parameters, and assuming the depth of the expected image is 150${\displaystyle mm}$ and the propagation speed of sound at 1500 ${\displaystyle m/s}$, each line will require 200 ${\displaystyle \mu s}$ to receive the back-scatter information from the farthest focal point. It can be seen that to satisfy the frame with 200 lines each, the frame rate of SAU can achieve a peak at 25 Hz, reflecting the appreciable potential in imaging speed and economy in implementation.^[3]

Virtual sources focusing

Passman and Ermert proposed using transmit focal points as virtual sources and this was further investigated in SAU by later researchers.^[10] This method treats the focus of a transducer as a virtual element besides the actual source points. SAU imaging can be performed without regard to whether or not the element actually exists, after the aperture angle of the transducer was determined. By introducing the virtual sources focusing method, the resolution of imaging and depth of penetration were increased. It was also suggested that the focusing in azimuth and elevation can be treated separately and that the concept of virtual sources focusing SAU can be applied in two planes.^[11][12]

Sequential beamforming

Kortbek has put forward a sequential beamforming method to reduce the complexity and improve the simplicity of the hardware implementation. The basic idea is to separate the beamforming process into two-stage procedures by using two independent beamformers. The beamformer in the first stage produces scan lines by using a single focal point in both the transmit and receive processes. The second stage beamformer creates a set of high resolution image points by combining information from multiple focused scan lines acquired in the first stage. For a linear array transducer with multiple elements, the lateral resolution of sequential beamforming SAU can be made more range independent and significantly improved compared to conventional dynamic transmit and receive focusing.^[13][14]

Bi-directional pixel-based focusing

The bi-directional pixel-based focusing (BiPBF) method was proposed to solve the problem that SAU imaging suffers from low SNR as the transmission is done by a small part of the array.^[15] In BiPBF, the same firing sequence is used in the transmission process as in the traditional array imaging, but the radio frequency (RF) data collected using adjacent groups of array elements are compounded in the receive process. The pixel-based time delays used for compounding are calculated using the distances between pixels and virtual sources located at the successive lateral positions of the transmission focus. Experiments have been done in both phantom and in-vivo experiments; the image quality of the SAU-BiPBF method was considerably improved compared to imaging by conventional dynamic focusing.^[16][17]